Partial Least Squares Regression on Symmetric Positive-Definite Matrices

被引:0
|
作者
Alberto Perez, Raul [1 ]
Gonzalez-Farias, Graciela [2 ]
机构
[1] Univ Nacl Colombia, Fac Ciencias, Escuela Estadist, Medellin, Colombia
[2] CIMAT Mexico Unidad Monterrey, Dept Probabilidad & Estadist, Monterrey, Nuevo Leon, Mexico
来源
REVISTA COLOMBIANA DE ESTADISTICA | 2013年 / 36卷 / 01期
关键词
Matrix theory; Multicollinearity; Regression; Riemann manifold; TENSOR; FRAMEWORK;
D O I
暂无
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
Recently there has been an increased interest in the analysis of different types of manifold-valued data, which include data from symmetric positive-definite matrices. In many studies of medical cerebral image analysis, a major concern is establishing the association among a set of covariates and the manifold-valued data, which are considered as responses for characterizing the shapes of certain subcortical structures and the differences between them. The manifold-valued data do not form a vector space, and thus, it is not adequate to apply classical statistical techniques directly, as certain operations on vector spaces are not defined in a general Riemannian manifold. In this article, an application of the partial least squares regression methodology is performed for a setting with a large number of covariates in a euclidean space and one or more responses in a curved manifold, called a Riemannian symmetric space. To apply such a technique, the Riemannian exponential map and the Riemannian logarithmic map are used on a set of symmetric positive-definite matrices, by which the data are transformed into a vector space, where classic statistical techniques can be applied. The methodology is evaluated using a set of simulated data, and the behavior of the technique is analyzed with respect to the principal component regression.
引用
收藏
页码:177 / 192
页数:16
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